The Bankruptcy Of Several Types Of Risk Models

This dissertation is devoted to dealing with the ruin theory for some kinds of risk models which include the ruin problem for a correlated aggregate claims model with Poisson and Erlang(2) risk processes, the ruin theory for a discrete-time risk model with dependence between claim sizes and the occurrence of claim and for the risk process perturbed by diffusion in a Markovian environment. In most actuarial literatures related to risk theory, the assumption of independence between classes of business in an insurance book of business is made. In practice, however, there are situations in which this assumption is not verified. Papers that treat a relation of dependence between classes of business include Ambagaspi-tiya(1998), Baurble and Muller(1998), Cossette and Marceau(2000), Wu and Yuen(2003), Yuen, Guo and Wu(2002), Albrecher and Boxma (2004) etc. Among them, Cossette and Marceau(2000) considered the Poisson model with common shock (PCS model) and the negative binomial model with common component (NBCC model), Wu and Yuen(2003) also considered a discrete-time risk model with interaction between classes of business (IR model), Yuen, Guo and Wu(2002) discussed a correlated aggregate claims model with Poisson and Erlang risk processes. Albrecher and Boxma (2004) considered a generalization of the classical ruin model to a dependent setting, where the distribution of the time between two claim occurrences depends on the previous claim size, and they derived the exact analytical expressions for the Laplace transform of the ruin function. In addition, because of the uncertainty of the payment or the income and the income of interest, so it leads to the risk model perturbed by diffusion and the risk model under interest and the Cox risk model. About the risk model under a constant interest, we can see Sundt and Teugels(1995), Sundt and Teugels(1997), Yang and Zhang(2001) etc. Sundt and Teugels(1995) dis-cussed infinite time ruin probabilities in continuous time in a compound Poisson process with a constant premium rate and a constant interest rate, and discussed equations for the ruin probability as well as approximations and upper and lower bounds. Sundt and Teugels(1997) continuously discussed the risk model defined by Sundt and Teugels(1995), and got more explicit information on the adjustment function. Yang and Zhang(2001) also considered the problem of the severity of ruin for the risk model defined by Sundt and Teugels(1995) and got the equations satisfied by the distributions of surplus immediately after ruin. There are many papers about the risk model perturbed by diffusion and the Cox risk model, we can see such references and the references therein as Dufresne and Ger-ber(1991), Gerber and Landry(1998), Tsai and Willmot(2002), Tsai(2003), Chiu and Yin(2003), Reinhard (ASTIN Bulletin XIV), Jasiulewicz(2001), Wang(2001), Wang and Wu(2000) and so on. Enlightened by these literatures, in this dissertation I'll consider two kinds of dependent risk model: one is a correlated risk model under constant interest, the other is a discrete-time risk model with dependence between claim sizes and the occurrence of claim, at last a Cox risk model perturbed by diffusion is discussed.The first chapter mainly discusses the expected value of a discounted penalty function at ruin of a correlated aggregate claims model with Poisson and Erlang (2) risk processes under a constant premium rate and a constant interest rate. The penalty is simply a function of the surplus immediately prior to ruin and the deficit at ruin. An integral equation for the expected value is derived. At the same time the equation for the ruin probability is derived, as well as the asymptotic property of the ruin probability for this risk process with general claim size distributions. In particular, the expression for the ruin probabilities under the assumed model is derived when the claim sizes are exponentially distributed.In chapter 2, a discrete-time risk model with dependence between claim sizes and the occurrence of claim is proposed. The model assumes that in...